# Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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### Direct proof for $\mathcal{O}_{\mathbb{CP}_{1}}(-2) \cong T^{*}\mathbb{CP}_{1}$

Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the ...
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### Polarization of abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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### Canonical bundle and canonical divisor in a $K3$ surface

The algebraic definition of a $K3$ surface is this: A smooth algebraic suface $X$ is called $K3$ if: i) $X$ has trivial canonical bundle; ii) $h^1(X,\mathcal{O}_X)=0$. I know that i) ...
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### Sections of very ample line bundle

Let $f: C \to D$ a dominant morphism which is not an isomorphism between two irreducible, reduced, projective curves $C,D$ over an alg closed field $k$ (unsure if algebraically closedness is ...
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### When is the image of a line bundle again a line bundle

Hello everybody Motivation of my question Let $X$ be a scheme. Given a morphism $\mathcal{L}\overset{\beta}\to\mathcal{O}_X$ of line bundles over $X$. I want to understand under what conditions the ...
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### Geometry of taking the sheaf of algebras associated to a line bundle.

Given a line bundle $L$ on a scheme $X$, we can construct the sheaf of $O_X$ algebras $\bigoplus\limits_{n=0}^\infty L^n$, which by the global Spec functor induces a map from some other scheme to $X$. ...
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### Are global sections of a line bundle canonically identified with rational functions while rational sections are not?

Every line bundle $L$ on a smooth complex projective variety $X$ is of the form $\mathcal O(D)$ for some divisor $D$ on $X$. The global sections of $\mathcal O(D)$ are identified with the set of ...
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### Ratio of sections of a line bundle

Does the ratio of two global sections of a complex line bundle on a compact Riemann surface define a rational function on this Riemann surface? Is the degree of this rational function (as a map from ...
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### Line bundles on projective space and disk

I'm having a difficult time solving some exercice. I should prove the following : Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic ...
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### Tautological bundle: algebraic geometry vs topology

I'm going to compare the two construction of twisted sheaf/bundle $\mathcal{O}(1)$ from algebraic and topological viewpoint: 1) Algebraic construction (Hartshorne's Algebraic Geometry, p. 117): ...
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### Is Wronskian a Line Bundle for Riemann surfaces?

Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...
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### Spin structure on a torus

Definition: A spin structure on a Riemannian manifold $M$ is a complex vector bundle S→M together with a isomorphism of algebra bundles Cl(M)→End(S). Here Cl(M) denotes the Clifford bundle over the ...
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### Does every real line bundle admit a flat connection?

Consider a three-fold intersection $U_{i} \cap U_{j} \cap U_{k}$ of a trivialising Leray cover $\{\mathcal{U}\}$ for a real line bundle L over a $C^k$ manifold $M$. We have $g_{ij}g_{jk}g_{ki} = 1$ ...
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### Global sections of square root line bundle

Let $C$ be a smooth curve in $\mathbb{P}^2$ over field $\mathbb{C}$. Suppose that I have a very ample line bundle $L$ on $C$ of even degree. Then $L$ has $2^{2g}$ square roots in $Pic\ C$. These are ...
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### Picard group of a conic

Suppose $C$ is a smooth conic in $\mathbb{P^2}$, that is, $C$ is a hypersurface of degree two in the projective plane. Assume that the base field is complex numbers. By the genus degree formula, we ...
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### Examples of non trivial vector bundles

Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles. Here, I want to collect such examples with justification of one or two lines saying ...
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A question about the definition of ample line bundle. On a Noetherian scheme, a line bundle $L$ is said to be ample if for any coherent sheaf $F$, there exists $n_0\in\mathbb N$ such that $F\otimes ... 0answers 89 views ### Closed embedding = very ample line bundle Let$\pi \colon X \to \mathbb{P}^n$be a closed embedding given via an invertible sheaf$\cal L$with global sections$s_0, \dots, s_n$. Thus${\cal L} \cong \pi^* {\cal O}_X$. Why is${\cal L} \...
Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact). Let $L$ be a line bundle over $B$. My question is how to see that the morphism ...
Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle \$...